$\frac{ 1 }{ 6 } \times \int{ \frac{ \sqrt{ {x}^{3} }-\sqrt[3]{x} }{ \sqrt[4]{x} } } \mathrm{d} x$
Simplify the radical expression$\frac{ 1 }{ 6 } \times \int{ \frac{ x\sqrt{ x }-\sqrt[3]{x} }{ \sqrt[4]{x} } } \mathrm{d} x$
Use $\sqrt[n]{{a}^{m}}={a}^{\frac{ m }{ n }}$ to transform the expression$\frac{ 1 }{ 6 } \times \int{ \frac{ x\sqrt{ x }-{x}^{\frac{ 1 }{ 3 }} }{ \sqrt[4]{x} } } \mathrm{d} x$
Use $\sqrt[n]{{a}^{m}}={a}^{\frac{ m }{ n }}$ to transform the expression$\frac{ 1 }{ 6 } \times \int{ \frac{ x\sqrt{ x }-{x}^{\frac{ 1 }{ 3 }} }{ {x}^{\frac{ 1 }{ 4 }} } } \mathrm{d} x$
Use $\sqrt[n]{{a}^{m}}={a}^{\frac{ m }{ n }}$ to transform the expression$\frac{ 1 }{ 6 } \times \int{ \frac{ x \times {x}^{\frac{ 1 }{ 2 }}-{x}^{\frac{ 1 }{ 3 }} }{ {x}^{\frac{ 1 }{ 4 }} } } \mathrm{d} x$
Calculate the product$\frac{ 1 }{ 6 } \times \int{ \frac{ {x}^{\frac{ 3 }{ 2 }}-{x}^{\frac{ 1 }{ 3 }} }{ {x}^{\frac{ 1 }{ 4 }} } } \mathrm{d} x$
Separate the fraction into $2$ fractions$\frac{ 1 }{ 6 } \times \int{ \frac{ {x}^{\frac{ 3 }{ 2 }} }{ {x}^{\frac{ 1 }{ 4 }} }-\frac{ {x}^{\frac{ 1 }{ 3 }} }{ {x}^{\frac{ 1 }{ 4 }} } } \mathrm{d} x$
Simplify the expression$\frac{ 1 }{ 6 } \times \int{ {x}^{\frac{ 5 }{ 4 }}-\frac{ {x}^{\frac{ 1 }{ 3 }} }{ {x}^{\frac{ 1 }{ 4 }} } } \mathrm{d} x$
Simplify the expression$\frac{ 1 }{ 6 } \times \int{ {x}^{\frac{ 5 }{ 4 }}-{x}^{\frac{ 1 }{ 12 }} } \mathrm{d} x$
Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$$\frac{ 1 }{ 6 } \times \left( \int{ {x}^{\frac{ 5 }{ 4 }} } \mathrm{d} x-\int{ {x}^{\frac{ 1 }{ 12 }} } \mathrm{d} x \right)$
Use $\begin{array} { l }\int{ {x}^{n} } \mathrm{d} x=\frac{ {x}^{n+1} }{ n+1 },& n≠-1\end{array}$ to evaluate the integral$\frac{ 1 }{ 6 } \times \left( \frac{ 4{x}^{2}\sqrt[4]{x} }{ 9 }-\int{ {x}^{\frac{ 1 }{ 12 }} } \mathrm{d} x \right)$
Use $\begin{array} { l }\int{ {x}^{n} } \mathrm{d} x=\frac{ {x}^{n+1} }{ n+1 },& n≠-1\end{array}$ to evaluate the integral$\frac{ 1 }{ 6 } \times \left( \frac{ 4{x}^{2}\sqrt[4]{x} }{ 9 }-\frac{ 12x\sqrt[12]{x} }{ 13 } \right)$
Distribute $\frac{ 1 }{ 6 }$ through the parentheses$\frac{ 2{x}^{2}\sqrt[4]{x} }{ 27 }-\frac{ 2x\sqrt[12]{x} }{ 13 }$
Add the constant of integration $C \in ℝ$$\begin{array} { l }\frac{ 2{x}^{2}\sqrt[4]{x} }{ 27 }-\frac{ 2x\sqrt[12]{x} }{ 13 }+C,& C \in ℝ\end{array}$